BABYL OPTIONS: -*- rmail -*- Version: 5 Labels: Note: This is the header of an rmail file. Note: If you are seeing it in rmail, Note: it means the file has no messages in it.  1, edited, forwarded,, Mail-from: From hovey@math.mit.edu Fri Feb 3 09:53:22 1995 Return-Path: Received: from nevanlinna.mit.edu by math.mit.edu (4.1/Math-2.0) id AA23567; Fri, 3 Feb 95 09:51:15 EST From: Mark Hovey Received: by nevanlinna.mit.edu; Fri, 3 Feb 95 09:51:11 EST Date: Fri, 3 Feb 95 09:51:11 EST Message-Id: <9502031451.AA21474@nevanlinna.mit.edu> To: hovey@math.mit.edu Subject: Hopf mailing list Reply-To: hovey@math.mit.edu *** EOOH *** Return-Path: From: Mark Hovey Date: May 13, 1995 09:51:11 EST To: hovey@math.mit.edu Subject: Hopf mailing list Reply-To: hovey@math.mit.edu This is the seventh installment of a mailing list of algebraic topology papers recently uploaded to Clarence Wilkerson's archive. Instructions at the end. Mark Hovey Papers uploaded to Hopf between May 3 and May 13, 1995: 1. /pub/Blanc/Blanc_hspace.abstract % % Homotopy operations and the obstructions to being an H-space % David Blanc % % November 14, 1994 % The question of whether a given space X possesses such an H-space structure has been studied from a variety of viewpoints. Here we address this question from the aspect of homotopy operations. This is done by reformulating the question in terms of the realizability of a certain morphism of abelian \Pi-algebras, which translates in turn (using the author's obstruction theory for realization of such morphisms) into the requirement that a certain sequence of higher homotopy operations, taking values in \pi_{\star} X, vanishes coherently. We illustrate the theory by a couple of examples: it can be used to calculate the obstruction to CP^2 being an H-space rationally; we also show that the "torsion Whitehead product" (which we define) may be thought of as ``the first higher order obstruction'' to being an H-space, and give another example. 2. /pub/Blanc/Blanc_loop.abstract % % Loop spaces and homotopy operations % David Blanc % % April 27, 1995 % We describe two obstruction theories for a given topological space X to be a loop space, both defined in terms of higher homotopy operations: First, we explain how an H-space structure on X can be used to define the action of the primary homotopy operations on the shifted homotopy groups \pi_{\star-1} X (which are isomorphic to \pi_{\star} Y if X\simeq\Omega Y). \ This action will behave properly with respect to composition of operations if X is homotopy-associative, and will lift to a topological action of the monoid of all maps between spheres if and only if X is a loop space. The obstructions to having such a topological action may be stated in terms of the author's obstruction theories for realizing Pi-algebras and their morphisms. A more concrete approach, which does not require a given H-space structure on X, yields the following: Theorem A: If X is a CW complex such that all Whitehead products vanish in \pi_{\star} X, then X is homotopy equivalent to a loop space if and only if a certain collection of higher homotopy operations vanish coherently. The higher homotopy operations in question depend only on maps between wedges of spheres, and take value in homotopy groups of spheres. They are constructed by means of a certain collection of convex polyhedra which may be of independent interest. 3. /pub/Blanc/model.abstract % % New Model Categories from Old % David Blanc % % (revision: January 25, 1995) % Model categories, first introduced by Quillen, have proved useful in a number of areas - most notably in his treatment of rational homotopy, and in defining homology and other derived functors in non-abelian categories. From a homotopy theorist's point of view, one interesting example of such non-abelian derived functors is the E^2-term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E^2-term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra. The original purpose of this note was to provide an element in this identification which appears to be missing from the literature: namely, an explicit model category structure for the category cCA of cosimplicial coalgebras as above. What one would really like is a model category for arbitrary categories of cosimplicial universal coalgebras, analogous to Quillen's treatment of simplicial universal algebras, which is based on Quillen's ``small object argument'', and implicitly uses a procedure for transfering model category structures by means of adjoint functors (in the direction of the left adjoint; the procedure is made explicit in the paper). Unfortunately, Quillen's procedure cannot be dualized, in the categorical sense. The reason is essentially set-theoretic: more can be said about maps into a sequential colimit of sets than about maps out of a sequential limit (and thus, for example, colim is exact, for R-modules, while lim is not). Therefore, for our purposes we describe alternative (and less elegant) conditions for using adjoint functors to create new model category structures. The dual version then allows us to define model category structures for certain categories of cosimplicial universal coalgebras - including cCA. ======================== 4. /pub/Blanc/towers.abstract % % Colimits for the Pro category of towers of simplicial sets % David Blanc % % January 18, 1995 % The Pro category of towers of spaces (and of other categories) has been studied in several contexts, and used for a variety of applications in homotopy theory, shape theory, geometric topology, and algebraic geometry - as well as in the study of v_n-periodicity in unstable homotopy theory. One problem in the usual version of the Pro category of towers is that, while finite limits and colimits exist, and may be constructed in a straightforward (levelwise) manner, the same does not hold for infinite colimits; and these were needed for the application to v_n-periodicity. The construction presented here embeds a suitable subcategory of the Pro category Tow of towers of simplicial sets in a certain category Net of strict Ind-towers, in which we have explicit constructions for all colimits, as well as finite limits. This category Net can thus be thought of as a cocompletion of the Pro category of towers of spaces. There are other cocomplete categories in which Tow may be embedded - for example, the category of all pro-simplicial sets, or the full category of all inductive systems of towers. One advantage of the approach described here is that one obtains a smaller, and more mangeable, cocompletion, in this special case, and the construction of the colimits may be made quite explicitly. A side effect of our approach is the elimination of certain ``phantom phenomena'' from the Pro category of towers. 5./pub/Elmendorf-Kriz-Mandell-May/ekmm.abstract Title: Rings, modules, and algebras in stable homotopy theory Authors: A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May address: Purdue University Calumet, Hammond IN 46323 email: aelmendo@math.purdue.edu address: The University of Michigan, Ann Arbor, MI 48109-1003 email: ikriz@math.lsa.umich.edu address: The University of Chicago, Chicago, IL 60637 email: mandell@math.uchicago.edu address: The University of Chicago, Chicago, IL 60637 email: may@math.uchicago.edu Let $S$ be the sphere spectrum. We construct an associative, commutative, and unital smash product in a complete and cocomplete category $\sM_S$ of ``$S$-modules'' whose derived category $\sD_S$ is equivalent to the classical stable homotopy category. This allows a simple and algebraically manageable definition of ``$S$-algebras'' and ``commutative $S$-algebras'' in terms of associative, or associative and commutative, products $R\sma_S R \darrow R$. These notions are essentially equivalent to the earlier notions of $A_\infty$ and $E_\infty$ ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of $R$-modules in terms of maps $R\sma_S M\darrow M$. When $R$ is commutative, the category $\sM_R$ of $R$-modules also has an associative, commutative, and unital smash product, and its derived category $\sD_R$ has properties just like the stable homotopy category. Working in the derived category $\sD_R$, we construct spectral sequences that specialize to give generalized universal coefficient and K\"{u}nneth spectral sequences. Classical torsion products and Ext groups are obtained by specializing our constructions to Eilenberg-Mac~Lane spectra and passing to homotopy groups, and the derived category of a discrete ring $R$ is equivalent to the derived category of its associated Eilenberg-Mac~Lane $S$-algebra. We also develop a homotopical theory of $R$-ring spectra in $\sD_R$, analogous to the classical theory of ring spectra in the stable homotopy category, and we use it to give new constructions as $MU$-ring spectra of a host of fundamentally important spectra whose earlier constructions were both more difficult and less precise. Working in the module category $\sM_R$, we show that the category of finite cell modules over an $S$-algebra $R$ gives rise to an associated algebraic $K$-theory spectrum $KR$. Specialized to the Eilenberg-Mac~Lane spectra of discrete rings, this recovers Quillen's algebraic $K$-theory of rings. Specialized to suspension spectra $\Sigma^{\infty}(\Omega X)_+$ of loop spaces, it recovers Waldhausen's algebraic $K$-theory of spaces. Replacing our ground ring $S$ by a commutative $S$-algebra $R$, we define $R$-algebras and commutative $R$-algebras in terms of maps $A\sma_R A\darrow A$, and we show that the categories of $R$-modules, $R$-algebras, and commutative $R$-algebras are all topological model categories. We use the model structures to study Bousfield localizations of $R$-modules and $R$-algebras. In particular, we prove that $KO$ and $KU$ are commutative $ko$ and $ku$-algebras and therefore commutative $S$-algebras. We define the topological Hochschild homology $R$-module $THH^R(A;M)$ of $A$ with coefficients in an $(A,A)$-bimodule $M$ and give spectral sequences for the calculation of its homotopy and homology groups. Again, classical Hochschild homology and cohomology groups are obtained by specializing the constructions to Eilenberg-Mac~Lane spectra and passing to homotopy groups. 6. /pub/Green-Leary/extra.abstract Chern classes and extraspecial groups David J. Green and Ian J. Leary djg@math.uchicago.edu leary@mpim-bonn.mpg.de Abstract: The mod-$p$ cohomology ring of the extraspecial $p$-group of exponent~$p$ is studied for odd~$p$. We investigate the subquotient~$ch(G)$ generated by Chern classes modulo the nilradical. The subring of~$ch(G)$ generated by Chern classes of one-dimensional representations was studied by Tezuka and Yagita. The subring generated by the Chern classes of the faithful irreducible representations is a polynomial algebra. We study the interplay between these two families of generators, and obtain some relations between them. ---------------------Instructions----------------------------- To subscribe or unsubscribe to this list, send a message to hovey@math.mit.edu with your e-mail address and name. Please make sure I am using the correct e-mail address for you. To get the papers listed above, point your WWW client (Mosaic, Netscape) to http://hopf.math.purdue.edu/pub/hopf.html. There are links to conference announcements, Purdue seminars, and other math related things on this page as well. You can also use ftp to hopf.math.purdue.edu, and login as ftp. Then cd to pub. Files are organized by author name, so papers by me are in pub/Hovey. If you want to download a file using ftp, you must type binary before you type get . To put a paper of yours on the archive, cd to /pub/incoming. Transfer the dvi file using binary, by first typing binary then put You should also transfer an abstract as well. To do this take the TeX file and save the abstract to a different file, without any \begin{document} commands or anything, and transfer that file. You can use ascii instead of binary for this. I am solely responsible for this mailing list---don't send complaints about it to Clarence. Thanks to Clarence for creating and maintaining the archive.